No, that’s just an urban myth.
A common misconception in popular culture is that the water in toilets or bathtubs drains in one direction in the Northern Hemisphere, and then drains the other direction in the Southern Hemisphere. They say it is the consequence of the Coriolis effect. For the earth the Coriolis effect would cause “moving objects on the surface of the Earth to appear to veer to the right in the northern hemisphere, and to the left in the southern.” This idea has been incorrectly perpetuated by several television programs, including an episode of The X-Files and one episode of The Simpsons. There have also been several science broadcasts and publications (including at least one college-level physics book) have made this incorrect statement, but it’s just not true.